Category:Dirichlet's Principle for Harmonic Functions

This category contains pages concerning Dirichlet's Principle for Harmonic Functions:

Let the function $\map u x$ be the particular solution to Poisson's equation:

$\Delta u + f = 0$

on a domain $\Omega$ of $\R^n$ with boundary condition:

$u = g$ on $\partial \Omega$

Then $u$ can be obtained as the minimizer of the Dirichlet's energy:

$\ds E \sqbrk {\map v x} = \int_\Omega \paren {\frac 1 2 \cmod {\nabla v}^2 - v f} \rd x$

amongst all twice differentiable functions $v$ such that $v = g$ on $\partial \Omega$ .

This result holds provided that there exists at least one function which makes the Dirichlet Integral finite.

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.